Visual Cryptography and Random Grids - Visual Cryptography and Secret Image Sharing - page 186

Note that this denition is dierent from that in Naor and Shamir's model.
Suppose that C 1 and C 2 are the two encoded shares produced by some (2,
2)-VCS using basis matrices M 0 and M 1 with respect to binary image B.
Then the security condition in (2, 2)-VCS is w(M b [1]) = (M b [2]) for b 2
f0, 1g where M b [i] is the ith row of M b for i 2 f1, 2g, while the contrast
condition becomes w(M 1 [1] or M 1 [2]) (M 0 [1] or M 0 [2]) > 0. Let c 1 in C 1
and c 2 in C 2 be the corresponding pixels of b in B. Let d = c 1 c 2 where d
is in D = C 1 C 2 . Let b(0) (b(1)) denote the pixel of b = 0(1) and d[b(0)]
(d[b(1)]) denote such d corresponding to b(0) (b(1)). Since each b = 0 (1) is
encoded according to M 0 (M 1 ), the contrast condition in (2, 2)-VCS assures
w(d[b(1)])w(d[b(0)]) > 1 so that d (1) can be identied from d (0) for all d(0)'s
and d(1)'s in D. However, VCRG only demands T (S[B(0)]) T (S[B(1)]) > 0,

or equivalently, t (s[b(0)]) t (s[b(1)]) > 0.

7.3.2 (2, 2)-VCRG Algorithms for Binary Images
Let random pixel (0, 1) be a function that returns a binary value 0 or 1 to
repr esent a transparent or opaque pixel, respectively, by a coin-flip procedure
and R 1 [i;j] denote the complement of R 1 [i;j]. Each of the following three al-
gorithms successfully encodes a secret binary image B into two random grids
R 1 and R 2 which constitute a set of (2, 2)-VCRG.

Algorithms 1{3. Sharing a binary image by two random grids

Input: A w h binary image B where B[i, j ] 2 f0, 1g (white or black),
16i6w and 16j6h
Output: Two shares of random grids R 1 and R 2 which reveal B when su-
perimposed where R k [i, j ] 2f0, 1g (transparent or opaque), 16i6w, 16j 6h
and k 2f1; 2g

Encryption(B)

Algorithm 1.

1. Generate R 1 as a random grid, (R 1 ) = 1=2

// for (each pixel R1[i, 1 [i;j], 16i6w, and 16j6h) do
//

T

R 1 [i;j] = random pixel (0, 1)

2. for (each pixel B[i, j ], 16i6w and 16j6h) do

2.1 if if (B[i, j ] = 0) then R1[i, 2 [i;j] = R 1 [i;j]

else R1[i, 2 [i;j] = R 1 [i;j]

g

3. output(R 1 , R 2 )